Sachen's name in Chinese is given as 聖謙, which in Pinyin would be Shèng qiān. So how does one arrive at the "Sachen" and "Thin Chen" readings? Or are they just made-up words to sound better as a company name?

Considering that Sachen is a Taiwan company, it is understandable to not use the more mainland Pinyin spelling.
Also, there is ABSOLUTELY no official standard (no matter what people tell you) for romanising Chinese characters, so whatever sounds close (and cool) is okay.

Actually in Mandarin Chinese (not otherwise actually, we non Mandarin speakers are actually baffled by this) sometimes treat sounds beginning with H and S interchangeable, and that H and F interchangeable, so one official Chinese translation for Sherlock Holmes sounds somewhat like "Halok Furmosi" to us (admittedly that this is more like how that would sound in Cantonese; if you read it in Mandarin it's closer to the original English).

Moreover, as shown in that wikipedia page you linked to, the company is actually more widely known as 富士康, which sounds VERY close to Foxconn.

Just on a related note, which is what's happening now here, that the content provider (and the big N itself) insisted on name changes to match the Mandarin version in Pokemon Sun+Moon (which sound awful unless pronounced in Mandarin, and this already happened in the games themselves), which caused the TV channel to rage quit and consider not to air the S+M anime after XYZ ends.